#219671

order	512
nilpotency class	3
isoclinism	isoclinic class of #219590
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3207	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,14)(4,18,22,27)(5,17)(6,12)(7,13)(8,10)(9,21,28,32)(11,23)(15,29,31,24)(16,25)(19,26,30,20)(33,34,39,43,37,42,38,54)(35,44,48,59,46,40,47,56)(36,49,53,58,51,63,52,41)(45,60,62,55,50,64,61,57);(1,3,6,10,7,17,23,16)(2,8,11,14,13,25,12,5)(4,19,32,15,22,30,21,31)(9,26,18,24,28,20,27,29)(33,35,38,47,37,46,39,48)(34,40,54,59,42,44,43,56)(36,50,52,62,51,45,53,61)(41,57,63,64,58,55,49,60);(1,4,7,22)(2,9)(3,15)(5,20,14,26)(6,21,23,32)(8,24,25,29)(10,19)(11,27)(12,18)(13,28)(16,30)(17,31)(33,36,39,53,37,51,38,52)(34,41,54,63,42,58,43,49)(35,45,48,62,46,50,47,61)(40,55,59,60,44,57,56,64);(1,5,23,25,7,14,6,8)(2,10,12,3,13,16,11,17)(4,20,32,29,22,26,21,24)(9,19,18,15,28,30,27,31)(34,42)(36,51)(40,44)(43,54)(45,50)(52,53)(56,59)(61,62)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=ehi,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dh,f^b=fh,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126867);